Numerical Analysis - Disconnected Contributions to Hadronic Processes

We shall now perform the numerical analysis to complete the procedure outlined in the previous section. We use the lattice QCD data ofππscattering provided to us by theEuropean Twisted Mass collaboration that uses N_f = 2+1+1 and lattice spacing a = 0.086 fm [76, 93]. We also use data from two different ensembles that have the same bare QCD parameters, but different volumes, corresponding to different lattice sizes. They are called A40.32 and A40.24, which correspond to (L/a)³×T/a = 32³×64 and 24³×48 respectively. As already elucidated, we are first required to extract the energy levels{E^α

0,E^β

0}using the connected (crossed (C) and direct (D)) correlation functions. We follow the technique described in [58], in order to minimise pollution from thermal states, and define,

R_i(τ+a/2) ≡ Ceⁱ(τ) −Ceⁱ(τ+a)

C_π²(τ) −C_π²(τ+a), (6.34) whereC_π(τ)is the single pion two-point correlation function andi=α, β. Next, we fit the data to a functional form of this function

R_i(τ+a/2)= A

cosh

δE₀ⁱτ⁰ +sinh

δE₀ⁱτ⁰

coth 2M_πτ⁰

, (6.35)

6.4 Numerical Analysis

0 5 10 15 20 25 30

1.5 1.6 1.7 1.8 1.9 2.0

τ/a Rα(τ+a/2)

0 5 10 15 20 25 30

2.0 2.1 2.2 2.3 2.4 2.5

τ/a Rβ(τ+a/2)

Figure 6.1: Fit to the functionR_i(τ+a/2)defined in Eq. (6.35) fori=α(top) andβ(bottom). The blue and red dots represent the data from the ensembles A40.24 and A40.32 of [76], respectively.

withτ⁰=τ+a/2−T/2 and the energy shiftδEⁱ

0is to be extracted, δE₀ⁱ =Eⁱ

0−2M_π. (6.36)

The pion masses used in [76] at finite volume areaM_π =0.1415(2)and 0.1446(3)for the ensembles A40.32 and A40.24 respectively. The relation of these finite volume values to the infinite volume values we require is outlined in AppendixD, along with the error analysis implemented in this fitting procedure.

We use the bootstrap method for estimating the errors of the fitted discrete energy levels [2]. In Fig.6.1, we plot the distribution of the mean values for 1500 bootstrap samples, wherein the error is taken as the standard deviation of this distribution. The accuracy of results from data fitting depends on the fit range, and it is preferable to consider a large number of fit ranges and average over the results using a carefully chosen weight [70]. Here, however, we compromise and perform a single fit for each ensemble over a specific fit range, and the best-fit curves are shown in Fig6.1. This fit range, 16a ≤τ ≤31aforA40.32 and 16a ≤τ ≤23aforA40.24, is chosen since it best reproduces theI =2 energy shifts in [70].

Chapter 6 Constraints on Disconnected Contributions inππScattering Correlation function aδE

0(A40.32) aδE

0(A40.24) a

0/a r

0/a

Ce^I=2 0.0033(1) 0.0082(3) 1.09(6) 53(107)

Ce^I⁼², SU(2) ChPT 1.300(19) 83(2)

Ce^α 0.0034(1) 0.0083(3) 1.124(54) 41(97)

Ce^β −0.0036(1) −0.0086(3) −1.429(77) 140(85)

Table 6.1: The fitted energy shifts, the extracted inverse scattering lengths and effective ranges (with an infinite volume pion mass of about 323 MeV) obtained using the connectedππcorrelation functions. For comparison, we list theI=2 values with the corresponding errors from the original lattice paper [76] in the second row, and include the NLO ChPT predictions of theI=2 threshold parameters in the third row.

The different fitted energy levels and the resulting threshold parameters are displayed in Table6.1.

It is evident that the scattering lengths are determined to a high accuracy, whereas the effective ranges are riddled with high errors. This conforms with our expectations since the effective range is the second term in the effective range expansion Eq. (6.18), and affects the discrete energy levels to a much smaller degree (Eq. (6.19)). This phenomenon has already been observed in [70]. We also quote the I =2 scattering length and effective range predicted by ChPT at NLO obtained from Eq. (6.30). There is a small discrepancy of about 2.4σbetween the scattering length of NLO ChPT and the performed fit to lattice data. This discrepancy probably arises due to the lattice fitting ofl^phys

1 andl^phys

2 , since these quantities are absent in the pion mass and pion-decay constant in infinite volume NLO ChPT, and are only estimated via finite size effects. Here, we introduce the ‘physical’ LECs ¯l_i^phy as,

l¯_i M_π

=l¯_i^phy −ln

M_π²/M_π,²

phy

, (6.37)

where the physical pion mass is,M_π,

phy≈138 MeV.

The next step of our process involves the fitting of the unphysical LECs n

L^PQ^,^r

0 ,L^PQ^,^r

by modifying the expressions fora^β

0 andr^β

0: 3L^PQ^,^r

0 +L^PQ^,^r

3 =− 1

512πM_πa^β

− F_π²

32M_π² 1+4µ_π + l

384π² + l

192π² − l₃

1024π² − l

256π² +L^PQ^,^r

5 −L^PQ^,^r

8 − 3

1024π² L^PQ^,^r

0 = 1

2048π − 3 M_πa^β

+M_πr^β

+ F_π²µ_π 6M_π² + l

384π² + l₃ 1024π² − 1

2 L^PQ^,^r

+L^PQ^,^r

8 + 17

6144π².

(6.38)

It is particularly beneficial to obtain a precise value of the linear combination 3L^PQ^,^r

0 +L^PQ^,^r

3 since it is independent of the effective ranger^β

0 and the large uncertainty that accompanies it, and it appears in the scattering lengths

n a^β

0,a^γ

. Our estimation of this linear combination agrees well with the NNLO fit mentioned in the previous chapter [84], and also has the advantage of being determined with a lower uncertainty. However, there is a disparity of about 1.6σin the values ofL^PQ

0 extracted by us and

6.4 Numerical Analysis Parameters Previous This Work

F0[MeV] 85.58(38) l¯^phy

1 −0.309(139) l¯^phy

2 4.325(10)

l¯^phy

3 3.537(47)

l¯^phy

4 4.735(17)

10³L^PQ^,r

5 0.501(43)

10³L^PQ^,r

8 0.581(22)

10³(3L^PQ^,r

0 +L^PQ^,r

3 ) −0.6(1.4) −0.70(18) 10³L^PQ^,r

0 1.0(1.1) 5.7(1.9)

Table 6.2: TheSU(2)ChPT andSU(4|2)PQChPT LECs relevant for our analysis. Values for the pion decay constantF₀in the chiral limit and the physical LECs{l¯_i^phy}are taken from [93]. The LECs{L^PQ,r

5 ,L^PQ,r

8 }and the combination{3L^PQ^,^r

0 +L^PQ^,r

3 ,L^PQ^,^r

0 }are obtained from [84] via NLO and NNLO fits respectively. In the third column, our fitted values of{3L^PQ^,^r

0 +L^PQ^,r

3 ,L^PQ^,^r

0 }are listed. The renormalisation scale of the unphysical LECs is 1 GeV.

the values obtained in [84]. Since our analysis has been entirely performed at NLO and the analysis in [84] required more unstable fits at NNLO, which are very sensitive to the fitting procedure, we assert that our result ofL^PQ

0 is more realistic and well-founded.

We now turn to the next step in our procedure - prediction of the energy shifts nδE^γ

0, δE₀^δo as functions of the lattice sizeL. We are now able to utilise our estimations of the LECs in Table6.2and calculate the scattering lengths and effective ranges in Eq. (6.32) and Eq. (6.33). Since the scattering lengtha^γ

0depends on the linear combination 3L^PQ^,^r

0 +L^PQ^,^r

3 , which has been determined reasonably precisely above, the value of the scattering length is estimated accurately. However, the expression for r^γ

0 containsL^PQ

0 that has a large uncertainty, particularly for the larger pion mass that increases effects from the counterterms. As mentioned already, the scattering length and effective range

n a^δ

0,r^δ

o for theI =0 amplitude depend only on physical LECs and are accurately estimated. The energy shift δE^δ

0 is then determined via a numerical solution of Eq. (6.19). The energy shifts are plotted in Fig.6.2 for 3< L M_π < 5, using three different pion masses, M_π =138, 236 and 330 MeV. These results warrant a few observations:

• Both theγ andδchannels have negative energy shifts, indicating that they are attractive in nature. Their magnitudes also increase with increasing pion mass and decreasing lattice size.

• The majority of the uncertainty ofδE^γ

0 is due tor^γ

0, which has a higher impact asδE^γ

0 gets larger. This indicates that the effective range expansion is breaking down in this limit.

• This leads to the error bar for δE^γ

0 diminishing even at a relatively large pion mass M_π of 330 MeV, withL M_π > 4, providing us with a controlled and accurate prediction of this energy shift.

We thus provide specific, concrete predictions of energy shifts that can act as consistency checks for lattice QCD calculations of disconnected diagrams in mesonic scattering.

Chapter 6 Constraints on Disconnected Contributions inππScattering

Mπ=138 MeV M_π=236 MeV Mπ=330 MeV

3.0 3.5 4.0 4.5 5.0

-0.08 -0.06 -0.04 -0.02 0.00

L M^π δE0γ (GeV)

Mπ=138 MeV Mπ=236 MeV Mπ=330 MeV

3.0 3.5 4.0 4.5 5.0

-0.08 -0.06 -0.04 -0.02 0.00

L M^π δE0δ (GeV)

Figure 6.2: Prediction of the energy shiftsδE^γ

0 andδE₀^δas a function of the lattice sizeLatM_π =138 MeV (blue), 236 MeV (red) and 330 MeV (purple).

Im Dokument Disconnected Contributions to Hadronic Processes (Seite 70-74)